Is there a crisis in senior police leadership? The prime minister seems to think so – and this was before the recent riots erupted across London and elsewhere.

David Cameron recently told the Commons that the system for producing top police leaders is too closed, with "too few – and arguably too similar, candidates applying for the top jobs". His preference seems to be for outsiders, like 63-year-old Bill Bratton, a former police chief in New York and Los Angeles, or "battle hardened Afghan colonels".

But are these problems representative of a deep-seated malaise, or something more specific?

Policy makers should, of course, await the outcome of the inquiry into phone-hacking being led by Lord Justice Levenson; the criminal investigation in Cleveland; and, possibly, a Scarman-like inquiry into the August 2011 riots before drawing any conclusions. But this is rarely the way in British politics. More typical is the sweeping generalisation and the attempt to seize the political high ground.

The question that should be asked is not whether there are specific problems of police leadership, but whether there is a systemic problem. The difference is illustrated by reference to the MPs' expenses scandal. If there had been no more than a handful of MPs who had misused the system, that would have been a specific problem, requiring specific action relating to the individuals. The misuse turned out to be systemic, requiring general reform.

Despite current rhetoric, there is no evidence of a systemic problem with senior police leadership. As with any organisation, individual performance varies but the evidence is that the police service handles most of the problems it is confronted with satisfactorily, and sometimes very well. Of course there are mistakes, but this does not amount to systemic failure. Mistakes, however, make headlines, and headlines often require a political response.

As for the response to the August riots, the police in England, Wales and Scotland are not structurally suited to transform to an anti-riot capability. All forces have specially trained officers with specialist equipment; senior officers trained in command and control, and well-exercised contingency call-out plans to concentrate reinforcements. But it still takes time to move from essentially "normal" policing to the large-scale deployments required to deal with the extraordinarily dynamic, widespread and destructive disorder of August 2011.

In these circumstances the test of leadership will be how quickly it adapts to the genuinely new circumstances.

In contrast to the specific criticism, the successes of the police service and its leaders often go unrecognised. In 2007 my chief officer colleagues and I led the multi-agency response to the Gloucestershire floods, one of the largest peacetime emergencies this country has seen.

It has generally been recognised as a model of its kind – but we received no word of formal thanks from representatives of central government. We did not expect any, but equally this, and other examples of sound crisis management, should not be ignored when assessing police leadership.

This does not mean that senior police leadership does not require timely and sensible development. Times change and leadership needs to change with them. More top graduates are needed and an officer cadet scheme is required.

In recent years government-led cost cutting has pared down strategic leadership training. It is time to reassert more broadly based strategic training, exposing leaders to higher levels of current academic thinking in economics, sociology, organisational theory and international police developments.

Finally, the government's elected commissioners will probably not appreciate strong professional leadership. It is time to institute a Home Office-led, centrally-selected cadre to ensure forces get the best available leaders.

The events of the past few days illustrate that the police have to deal with a social, legal and organisational issue of great complexity and sensitivity. It is no place for amateurs, no matter how gifted.

Dr Tim Brain is the former chief constable of Gloucestershire and is an honorary senior research fellow at Cardiff University

This article is published by Guardian Professional. Join the Guardian Public Leaders Network to receive regular emails on the issues at the top of the professional agenda.